We show that unless $\NP \subseteq \RTIME (2^{\poly(\log{n})})$,
there is no polynomial-time algorithm approximating the Shortest
Vector Problem ($\SVP$) on $n$-dimensional lattices in the $\ell_p$
norm ($1 \leq p0$. This improves the
previous best factor of $2^{(\log{n})^{1/2-\eps}}$ under the same
complexity assumption due to Khot (J. ACM, 2005). Under the
stronger assumption $\NP \nsubseteq \RSUBEXP$, we obtain a hardness
factor of $n^{c/\log\log{n}}$ for some $c>0$.

Our proof starts with Khot's $\SVP$ instances that are hard to
approximate to within some constant. To boost the hardness factor
we simply apply the standard tensor product of lattices. The main
novel part is in the analysis, where we show that the lattices of
Khot behave nicely under tensorization. At the heart of the
analysis is a certain matrix inequality which was first used in the
context of lattices by de Shalit and Parzanchevski.